Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-18T11:30:21.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Ensembles invariants des flots géodésiques des variétés localement symétriques

Published online by Cambridge University Press:  19 September 2008

A. Zeghib
A. Zeghib
Affiliation:
CNRS UMR 128, Ecole Normale Supérieure de Lyon, 46, Allée d' Italie, 69364 LYON Cedex 07, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 2 , April 1995 , pp. 379 - 412
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ano]Anosov, D. V.. Geodesic flows on compact manifolds of negative curvature. Trud. Math. Inst. Steklova 90 (1967).Google Scholar
[BGS]Ballmann, W., Gromov, M. and Schroeder, V.. Manifolds of Non-positive Curvature. Progress in Mathematics. Vol. 61. Birkhauser: Boston-Basel-Stuttgart, 1985.CrossRefGoogle Scholar
[BK]Buser, P. and Karcher, H.. Gromov's almost flat manifolds. Astérisque 81 (1981).Google Scholar
[Fed]Federer, H.. Geometric Measure Theory. Springer: Berlin, 1969.Google Scholar
[Fra]Franks, J.. Invariant sets of hyperbolic toral automorphisms. Amer. J. Math. 99 (1977), 1089–1095.Google Scholar
[Ghy]Ghys, E.. Dynamique des flots unipotents sur les espaces homogènes. Sém. Bourbaki. Exposé 747 (1991).Google Scholar
[Han]Hancock, S.. Construction of invariant sets for Anosov diffeomorphisms. J. London. Math. Soc. 18 (1978), 339–348.Google Scholar
[Hel]Helgason, S.. Differential Geometry, Lie Groups and Symmetric Spaces. Academic: New York, 1978.Google Scholar
[Hir]Hirsch, M.. On invariant subsets of hyperbolic sets. Essays in Topology and Related Topics. (1970), 126–146.CrossRefGoogle Scholar
[Man1]Mañé, R.. Orbits of paths under toral automorphisms. Proc. Amer. Soc. 73 (1979), 121–125.CrossRefGoogle Scholar
[Man2]Mañé, R.. Invariant sets of Anosov diffeomorphisms. Invent. Math. 46 (1978), 147–152.Google Scholar
[Mau]Mautner, F. I.. Geodesic flows on symmetric Riemannian spaces. Ann. Math. 65 (1957), 416–431.CrossRefGoogle Scholar
[Mol]Molino, P.. Riemannian Foliations. Birkhauser: Boston—Basel—Stuttgart, 1988.CrossRefGoogle Scholar
[Mor]Morgan, F.. Geometric Measure Theory. Academic: New York, 1987.Google Scholar
[Mos1]Mostow, G.. Some new decomposition theorems for semisimple groups. Mem. Amer. Math. Soc. 14 (1961), 31–54.Google Scholar
[Mos2]Mostow, G.. Strong rigidity of symmetric spaces. Ann. Math. Studies. 78 (1973).Google Scholar
[Rag]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Springer: Berlin, 1972.Google Scholar
[Sta]Starkov, A. N.. Structure of orbits of homogeneous flows and the Raghunathan conjecture. Russian Math. Surveys 45 (1990), 227–228.CrossRefGoogle Scholar
[War]Warner, G.. Harmonic Analysis on Semi-simple Lie Groups I. Springer: Berlin, 1972.Google Scholar
[Wol]Wolf, J.. Spaces of Constant Curvature. Publish or Perish: New York, 1967.Google Scholar
[Zeg1]Zeghib, A.. Feuilletages géodésiques des varietés localement symétriques et applications. Thése. Dijon, 1985.Google Scholar
[Zeg2]Zeghib, A.. Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques. Ergod. Th. & Dynam. Sys. 14 (1994), 175–207.Google Scholar